If `f(x)= x^2+7x+10` then `f'(2) =? `

Options

A. -4
B. `-5/2`
C. -11
D. 11

Answer

To find the value of \( f'(2) \) for the function \( f(x) = x^2 + 7x + 10 \), we will follow these steps:

### Step 1: Find the derivative \( f'(x) \)
To find the derivative of the function \( f(x) = x^2 + 7x + 10 \), we will use the power rule of differentiation. The power rule states that if \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \).

- The derivative of \( x^2 \) is \( 2x^{2-1} = 2x \).
- The derivative of \( 7x \) is \( 7 \cdot x^{1-1} = 7 \).
- The derivative of the constant \( 10 \) is \( 0 \).

Combining these results, we have:
\[ f'(x) = 2x + 7 \]

### Step 2: Substitute \( x = 2 \) into \( f'(x) \)
Now that we have \( f'(x) = 2x + 7 \), we will substitute \( x = 2 \) to find \( f'(2) \):
\[ f'(2) = 2(2) + 7 \]

### Step 3: Calculate \( f'(2) \)
Now we will perform the calculation:
\[ f'(2) = 4 + 7 = 11 \]

### Final Answer
Thus, the value of \( f'(2) \) is \( 11 \).

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